We handle first the general lower bound. For each positive integer \(i\) , take \(W\)  to be the weighted graph defined as follows.
Take a clique of size \(\left\lfloor n^{\frac{t}{t+1}}\right\rfloor \)  and connect one of its vertices to \(n\)  additional vertices. Then \(t\left(K_{1,t},W\right)\)  is \(\Theta \left(n\right)\) , while \(t\left(G,W\right)=\Theta \left(n^{\frac{t+1}{|V\left(G\right)|-\delta _t\left(G\right)}}\right)\) ; hence, this gives the desired lower bound.
Next, we show that equality in fact holds when \(t=1\) . For this, we use the following two lemmas from [1]}.
Lemma 3.4 Let \(H\)  be any graph with \(\delta _1\left(G\right)=0\) . Then \(H\)  can be vertex-partitioned into edges and odd cycles.

Lemma 3.5 For any graph \(G\) , there exists an independent set \(I\)  with \(|I|-|N\left(I\right)|=\delta _1\left(G\right)\) .

Now, take any graph \(G\) , and let \(I\)  be an independent set chosen such that \(|I|-|N\left(I\right)|=\delta _1\left(G\right)\) . Let \(H\)  be the subgraph of \(G\)  obtained by removing \(I, N\left(I\right)\)  from \(G\) . We first claim that \(\delta _1\left(H\right)=0\) . Indeed, suppose otherwise, and let \(S\subset V\left(H\right)\)  be a subset of vertices with \(|S|>|N\left(S\right)|\) . Then
\(|I\cup T|-|N\left(I\cup T\right)|=|I|+|T|-N\left(I\right)-N_H\left(T\right)>\delta _1\left(G\right),\) 
a contradiction to the definition of \(\delta _1\) .
Now, let \(J\)  denote the subgraph of \(G\)  defined by removing \(H\) , so that \(J\cup H\)  is a subgraph of \(G\) . We claim that every vertex in \(N\left(I\right)\)  can be matched to a unique vertex in \(I\) , so that the resulting edges are disjoint. Indeed, for this we need only check the Hall's Marriage lemma condition, that for any subset \(R\subset N\left(I\right)\) , we have \(|N_J\left(R\right)|\ge |R|\) . Suppose otherwise, that some \(R\)  satisfies the opposite inequality. Then
\(|I\setminus N_J\left(R\right)|-|N\left(I\setminus N_J\left(R\right)\right)|&= |I|-|N_J\left(R\right)|-|N\left(I\right)|+2\left(|N\left(I\right)\setminus N\left(I\setminus N_J\left(R\right)\right)|\right)\\ &\ge \delta _1\left(G\right)+|R|-|N_J\left(R\right)|\\ &>\delta _1\left(G\right),\) 

once again a contradiction. Therefore such a matching indeed exists, and it follows that:
\(\rho \left(G, \begin{tikzpicture}[scale=0.3,auto=left,every node/.style={circle, draw=black, fill=black, minimum size=1pt,inner sep=1pt}]\node \left(n1\right) at \left(0,0\right) {};\node \left(n3\right) at \left(2,0\right) {};/\rightarrow in {n1/n3}\left(\right) -- \left(\rightarrow \right);\end{tikzpicture}\right)\le \frac{1}{\frac{|V\left(H\right)|}{2}+|J|}=\frac{2}{|V\left(G\right)|+|J|-|I|}=\frac{2}{|V\left(G\right)|-\delta _1\left(G\right)}\) 
as desired.
